Transmit beamforming can enhance the performance of multiple-input multiple-output (MIMO) systems by exploiting channel state information (CSI) at the transmitter. In the frequency-division duplex (FDD) mode, where the downlink and uplink channels are not reciprocal, the receiver must feedback information about the downlink channel to the transmitter. In systems with many transmit antennas, the feedback overhead can be overwhelming; and the challenge is to limit the feedback to only a few bits that still provide sufficient information about the channel.
Current technology may utilize transmit beamforming with limited feedback to address this challenge by designing efficient beamformer weight vector quantization algorithms at the receiver. In some implementations, the focus is on designing a common beamformer codebook the is known both at the transmitter and receiver. At runtime, the receiver estimates the downlink channel, finds the best-matching beamforming vector in the codebook, and feeds back a corresponding index into the codebook to the transmitter. Codebook design can be based on maximizing the average signal-to-noise ratio (SNR), maximizing the average mutual information, or minimizing the outage probability, and it can be viewed as a vector quantization problem, where the generalized Lloyd algorithm (GLA) can be used to construct the codebook. This codebook-based framework assumes accurate CSI at the receiver, which in turn implies significant downlink pilot overhead. For large codebooks, which are necessary when the number of transmit-antennas is large, the feedback overhead can be significant, and the computational complexity of searching the codebook for the best beamformer can be prohibitive.
Another issue is that assumption of a Rayleigh block-fading model, according to which the channel remains constant over a block of symbols and changes independently across different blocks. The block-fading assumption overlooks the channel temporal correlation, which can be exploited to decrease the feedback rate. The temporal correlation of the channel can be exploited by modeling the quantized CSI at the receiver as a finite-state Markov chain, and computing the transition probability of every codebook entry given the previous (one or more) codebook entries. As one example, variable-length Huffman source coding is applied to the transition probabilities of the Markov chain to compress the CSI feedback. This approach is not suitable for practical communication systems with limited feedback, which provision a fixed number of feedback bits per CSI slot, as in e.g., LTE. As another example, a different fixed-length but lossy CSI compression algorithm can be used, where low-probability transitions between the Markov chain states are truncated. For large-size codebooks, computing the transition probabilities accurately for a large number of Markov states is an elusive task that requires very long training periods. Moreover, the transition probabilities are dependent on the specific channel model new computations are necessary whenever the model varies significantly.